Articles of measure theory

Weak convergence and integrals

Assume $$u_k\rightharpoonup u,\quad v_k\rightharpoonup v\quad\text{in}\quad L^1(0,T;Y)\tag{1}$$ and $$\int_0^T u_k(t)\varphi'(t)\ dt=-\int_0^T v_k(t)\varphi(t)\ dt\tag{2}$$ for some $\varphi\in C_0^\infty(0,T)$. “Passing to the limit” we get $$\int_0^T u(t)\varphi'(t)\ dt=-\int_0^T v(t)\varphi(t)\ dt.\tag{3}$$ My question is: How to justify this passage to the limit? Remark: this passage to the limit is a step of the proof that “generalized derivatives are compatible […]

How to derive a union of sets as a disjoint union?

$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$ The results is obvious enough, but how to prove this

Are open sets in $\mathbb{R}^2$ countable unions of disjoint open rectangles?

I’m just trying to prove that each open set in $\mathbb{R}^2$ is measurable using the product measure derived from the Lebesgue measure. This is the first thing that came to mind because I recall that open sets are countable unions of disjoint open intervals, but I was wondering if there is an elementary way to […]

$B$ is a Borel set, implies $f(B)$ is a Borel set.

I have this exercise: Suppose $f$ is a real valued continuous function on $[a,b]$. Show that $f(B)$ is a Borel set for every Borel subset $B$ of $[a,b]$. Hint: Consider the collection $M$ of all subsets $A$ of $[a,b]$ for which $f(A)$ is a Borel set. Show that $M$ is a sigma-algebra. I am struggling […]

Is a Sobolev function absolutely continuous with respect to a.e.segment of line?

Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes. It seems to me that there is nothing special about the coordinate axes, so indeed if some direction $v\in \partial B(0,1)$ is fixed, we must have […]

Limit of $L_p$ norm as $ p \rightarrow 0$

I have reviewed Ayman Houreih’s proof for the limit of the $L_p$ norm as $ p \rightarrow 0$ at "Scaled $L^p$ norm" and geometric mean. While I have found the outline of the proof very helpful, I have a question regarding the applicability of LDCT: The proof mentions $\frac{|f|^q – 1}{q}$ as the dominating function. […]

If $f_{n}$ are non-negative and $\int_{X}f_{n}d\mu=1$ does $\frac{1}{n}f_{n}$ converge almost-everywhere to $0$? does $\frac{1}{n^{2}}f_{n}$?

Question from an exam sample I’m studying for: Suppose $\left(X,\mathcal{F},\mu\right)$ is a measure space and $f_{n}:\left(X,\mathcal{F}\right)\to\mathbb{R}$ a sequence of non-negative integrable functions such that $\int_{X}f_{n}d\mu=1$ . Is is it necessarily true that $\frac{1}{n}f_{n}$ converges almost everywhere to $0$ ? what about $\frac{1}{n^{2}}f_{n}$ ? My line of thought: From Fatou’s llema we got that: $$\int\limits _{X}\liminf_{n\to\infty}\frac{1}{n}f_{n}d\mu\geq\liminf_{n\to\infty}\int\limits […]

Diffuse-like decomposition of the segment $$ in accordance with Lebesgue measure

Consider the segment $[0,1]\subset\mathbb{R}$ and the standard Lebesgue measure $\mu$ on $\mathbb{R}$. I wonder if we can find such decomposition $A\sqcup B=[0,1]$, that for any subsegment $[a,b]\subset[0,1]$ we’d have $\mu(A\cap [a,b])=\mu(B\cap [a,b])$? More particular case is that for any $x\in[0,1]$ we have $\mu(A\cap [0,x])=\mu(B\cap [0,x])$ and hence $\mu(A\cap [0,x])=\frac{x}{2}$. Metaphor. Such decomposition can be assosiated […]

Measure Theory – Absolute Continuity

A question from my homework: Suppose $\mu$ is a measure on the real line w.r.t to the Borel $\sigma$-algebra such that $\forall x \in \mathbb{R}$ $\mu(\{x\})=0 $. is $\mu$ necessarily absolutely continuous w.r.t. to the Lebesgue measure? We say $\mu$ is absolutely continuous w.r.t. to $m$ if for every measurable set $E$ such that $m(E)=0$ […]

Existence of strictly positive probability measures

Let $X$ be a Hausdorff space (or let’s even assume it is metrizable). A strictly positive measure on $X$ assigns positive measure to any non-empty open subset of $X$. What conditions on $X$ ensure the existence of a strictly positive probability (or equivalently $\sigma$-finite) measure? Is there a standard way to construct such a measure […]